8 red balls

Probability Level 5

You have a bag with 8 red balls in it.

Every turn you randomly select a ball from the bag. If it is red you replace it with a blue ball, and if it is blue you replace it with a red ball.

What is the expected number of turns you need to make before all 8 balls in the bag are blue?

If the answer is $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers , what is $a+b$ ?

1 solution

Geoff Pilling
Jul 30, 2016

To solve this you need to solve the following set of linear equations:

$E_0 = 1 + E_1$
$E_1 = 1 + (1/8)*E_0 + (7/8)*E_2$
$E_2 = 1 + (2/8)*E_1 + (6/8)*E_3$
$E_3 = 1 + (3/8)*E_2 + (5/8)*E_4$
$E_4 = 1 + (4/8)*E_3 + (4/8)*E_5$
$E_5 = 1 + (5/8)*E_4 + (3/8)*E_6$
$E_6 = 1 + (6/8)*E_5 + (2/8)*E_7$
$E_7 = 1 + (7/8)*E_6$

where $E_n$ represents the expected number of moves given that you have $n$ blue balls in the bag.

Solving this for $E_0$ gives, $E_0 = 32768/105$ .

$32768 + 105 = \boxed{32873}$